Math July 13, 2026 · 8 Min Read

Absolute Value Equation Calculator – Guide & Formulas

Solve absolute value equations instantly. Enter any equation with |x| and get step-by-step solutions for both positive and negative cases.

Solve absolute value equations instantly with our free calculator. Enter any equation involving |x| and get both solutions with clear step-by-step breakdowns.

Key Takeaway

Use the free Absolute Value Equation Calculator to solve absolute value equations instantly. enter any equation with |x| and get step-by-step solutions for both positive and negative cases. Get instant results with step-by-step explanations.

How to Use the Absolute Value Equation Calculator

  1. Enter the absolute value expression on the left side (e.g., 2x + 3).
  2. Enter the value on the right side of the equation.
  3. Choose the equation type: |expression| = value or |expression| = expression.
  4. Review both solutions and the detailed step-by-step solving process.

The Formula

If |expression| = value (where value >= 0), then expression = value OR expression = -value. Solve each case separately. If value < 0, no solution exists.

Variable Definitions

  • |x|: The absolute value of x — distance from zero on the number line
  • a: The non-negative value on the right side of the equation
  • x: The unknown variable to solve for

Solving |2x - 5| = 7

Find x when the absolute value of (2x - 5) equals 7.

  1. Step 1: Set up two cases: 2x - 5 = 7 OR 2x - 5 = -7.
  2. Step 2: Case 1: 2x - 5 = 7 → 2x = 12 → x = 6.
  3. Step 3: Case 2: 2x - 5 = -7 → 2x = -2 → x = -1.
  4. Step 4: Both solutions are valid. Answer: x = 6 or x = -1.

Frequently Asked Questions

What is an absolute value equation?

An absolute value equation contains the absolute value of a variable expression, such as |x - 3| = 5. The absolute value represents the distance from zero, so you must consider both positive and negative cases.

Why does an absolute value equation have two solutions?

Since |a| = |-a| for any number, an equation like |x| = 7 has two solutions: x = 7 and x = -7, because both are the same distance from zero on the number line.

What if the right side is negative?

If |expression| = negative number, there is no solution. Absolute value is always non-negative (zero or positive), so it can never equal a negative number.

Can an absolute value equation have one solution?

Yes, if the right side equals zero (e.g., |x - 3| = 0), there is exactly one solution: x = 3. Zero is the only number whose absolute value is itself.

How do I check my solutions?

Substitute each solution back into the original equation. For |2x - 5| = 7: if x = 6, |12 - 5| = |7| = 7. If x = -1, |-2 - 5| = |-7| = 7. Both check out.